Is 0^0 1(pro) or undefined(con)?
round 1:acception and reasoning
round 2,3:main arguments
round 4: rebuttal
0^0 is undefined
I accept this debate.
I will be arguing upon the behalf of 0^0 = 1. The BoP is SHARED, as the Instigator says that 0^0 = undefined. So Con has to prove 0^0 = undefined and I have to prove 0^0 = 1.
If neither of us successfully prove their statement, voters, please feel free to vote on other topics such as s/g and citations.
I will expand upon my belief further. Best of luck.
First thanks for joining the debate.
anything to the power of zero like 3^0 is the same as saying 3/3
so 0^0 is the same as saying 0/0. anything divided by zero is undefined
Thank you for your argument. Keep in mind I won't rebut your statements this round is customary to be an opening argument.
First of all, I'd like to point out that if you type this into a calculator, sometimes, you'll get a "domain error". Some display 1. So it depends on the calculator as to what the answer is. But I shall argue for my case.
(The below is paraphrasing from a site I shall link.)
To start off this debate, I shall introduce a machine dubbed: "The Expander". Cheesy, I know, but I couldn't think of a better name.
Anyway, what The Expander does is (as the name says) takes my number and expands it!
For instance, let's say I wanted to change 1 to 16. I would put it onto 4x growth setting for two seconds. After it comes out, I have 16! I just preformed 4^2, which equals 16.
I can do this to ANY number. If I were to put 18 in the Expander for 2 seconds, I would get 324! If I were to put 5 in the machine for 3 seconds, I would get 125!
Now, what would happen if I were to want to put 7 in the machine for zero seconds?
Seems odd, right?
But the truth is, nothing happens then. I put 7 in the machine, take it out, it would be the same. The new number would be the same as the old number, therefore, the scaling factor of this would be 1.
Before I go further, let me explain scaling factors in the terms of this debate.
Scaling factors in multiplication can be considered easy. Simply type 3*4 = 12. 4 is the scaling factor, as I want three four times larger.
However exponential scaling factors can be considered strange in comparison to this. The Expander doesn't really know how it'll end up. As the amount of exponential growth increases, as does the number. 2^1 = 2. 2^2 = 4. 2^3 = 8. 2^ 4 = 16. There is no "set" growth. The more amount of seconds is put into the Expander, the more it'll grow...exponentially.
This applies to any other number, whether it be 5, 22, 9001 (heh), and yes, even 0.
If I were to put nothing in the machine for 0 seconds, nothing happens. But the new number would equal the old number. Therefore, the scaling would equal one, and that's how 0^0=1!
I look forward to R3's arguments.
Any other number to the power of 0 is 1 (that's the Zero Exponent Property). On the other hand, 0 to the power of anything else is 0, because no matter how many times you multiply nothing by nothing, you still have nothing.
So let's use one of the other properties of exponents to solve the dilemma:
Product of Powers Property
ab × ac = a(b + c)
Let's let a = 0, b = 2, and c = 0. Substituting, we have:
0^2 × 0^0 = 0(2 + 0) = 02
We know that 02 = 0. So this says
0 × 00 = 0
Notice that 00 can be equal to 0, or 1, or 7, or 99,999,999,999, and this equation will still be true!
For this reason, mathematicians say that 00 is undefined.
Sadly, I'd like to point out that opponent's argument is DIRECTLY copied from another source. You can see the said source here:
This is directly where my opponent got his arguments, with no modification to them whatsoever.
Opponent should have at least paraphrased said article to his own words, yet he didn't seem to say he did, let alone cite the article/page he got it from. Because of this, opponent seems to imply this is his own words (even though it's not), which is considered plagiarism. I ask the voters to penalize Con for his misuse of resources available.
Nonetheless, I shall refute.
First of all, let me state opponent's conclusion (or the conclusion he copied):
"For this reason, mathematicians say that 0^0 is undefined."
In my first argument, I said that in some instances when you input 0^0 into calculators, some will give an error, and some will give 1. It depends on the calucator and the mathematician to determine what the answer is. In my source, I give a reason as to why 0^0 = 1.
And to refute the point opponent stated: 0 x 0^0 = 0...this is just true fact. If anything was multiplied by 0, it equals zero, including numbers such as 12, 17, and one (which 0^0 power is, or at least I'm convinced).
This leads into my own reasoning, so I might as well continue here.
Back to the Expander argument.
If I put 7 in the machine for 0 seconds, nothing happens to the number. No matter what growth is set, if I have the setting on for 0 seconds, and press "Start", nothing will happen, therefore the scaling factor would be one. 7^0 = 1.
The same applies to 0. If nothing was put into the machine for 0 seconds, does it grow or shrink? Neither. It stays the same. Therefore the scaling factor would be one. Therefore 0^0 = 1.
Second, assume x is a variable. Taking the statement x^0 power and defining it as an empty product results in the answer being one. This is true for any number. x = 21, x= 43, and yes, even x = 0. Therefore 0^0 = 1.
Third, tuples. Tuples are ordered lists of elements. Combinational interpretation is the number of sets there are in a tuple set. In x^0, there would be one 0-tuple set. This can apply to any number, zero included. Therefore 0^0 = 1.
In addition with the above, 0^0 = 1 helps a lot of theorems in advanced math to run smoothly.
I accepted this debate to see what Con would say about this topic. Instead, he copies all of information from other sites. I'm going to quote a cite for the voters (and Con) to see. Refer to this example:
This explains 0^0 = 1 more than I ever could.
Read all the way to the bottom, and not just the top part (as it favors Con somewhat).
If you read it all the way through, you will notice it explains WHY people prefer to say 0^0 = 1. But in the end, 0^0 is defined as 1 to help out a lot of the theorems we know today and to make them simpler.
Looking forward to closing arguments, as long as you don't plagiarize them. :)
volcan forfeited this round.
Opponent concedes round four. Extend all arguments.
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